Desperate! Daily Backup in Bathtub (pics) Please Help Identify the Cause or Debris!

Hi, and thanks in advance for your help.
Please let me give you all the facts, then the description.

I live in a 50 year old apartment building in Brooklyn, NY.
My family has been this apartments only tenant, the entire time.

Though not a "luxury," building, tenants are paying from $2,000 to $4,000 for apartments here.
So it's supposed to be a nice place...

Suddenly, just in the last few weeks, the bathtub is backing up every morning, leaving a nasty residue rising up to six inches around the tub.
It backs up even if I close the external valve, (pictured).

I have to spend twenty minutes scrubbing this nasty residue, because it is so extremely difficult to clean off.
(Doesn't everyone spend the first twenty minutes of every day scrubbing their tub so they can then shower and catch the 7:15 am express bus?!)

The handymen here have snaked it, and it makes no difference.

I am including pictures of this detritus, which as you can see is reddish, and sticks stubbornly to the tubs surface.

What's worse is I have a new roommate moving in for September, and I imagine if this is not cleared up promptly, I could lose the tenant...

Has anyone ever seen this sort of debris before, and can
1. identify it's possible causes, therefore possible solutions?
2. identify what this debris is, or might be?

As snaking the line was totally ineffective, could it be the drains, air vent is clogged? (I'm assuming drains have air vents...)

Thanks again for any and all your help, and suggestions.
This is becoming a huge problem, and now may very well cost me, not only real money, but worse losing a roommate I have spent a lot of time interviewing for.
Heck, I wouldn't move in if the tub looked like that every morning.

There's a blockage in the drain somewhere below (or in) your unit. A blocked vent won't cause that issue. If it didn't come up in the tub, it would just find the next place which might be the toilet or a sink and since they're not as large, it might overflow onto the floor. The drain cleaning job didn't go far enough, or wasn't done well.

If you were to run water long enough, your tub and likely other things (sinks, toilets, etc) would back up, too.

The drains on some tubs rely on the pressure of the water in the tub to help push the rubber seal onto its seat, keeping the water in. Pressure from below is likely overcoming that seal and allowing it to push into the tub. To be blunt...it's likely the highly diluted sewage of those above you that can't get out to the main sewer line because there's some blockage, slowing it down. This is more likely to happen when lots of people are using their showers, or laundry, or whatever at the same time, creating max flow that your sewer line can no longer handle. If your building is at a low point along the roadway, it's possible that the whole sewer line isn't that deep, and if there's a blockage there it might show up at a low point in your building, but that would likely be noticed by the city.

Hope I'm wrong...maybe some others will have some better possibilities, but I still thing it's related to a blockage somewhere that must be addressed.

"To be blunt...it's likely the highly diluted sewage of those above you that can't get out to the main sewer line because there's some blockage, slowing it down."

As to detritus I then see every morning, as pictured above, would you say that this sewage has left behind rust that came off the drain pipes liner.
Have you seen that before?
Do rust flakes just occur in the sewage / drainage water, without needing any particulate matter to attach themselves to? For instance I never see hair in there.

I have to believe the answer is getting a real plumber with professional equipment to clean the drain. Now, the first, and perhaps greatest problem, is convincing the building owner to hire someone beside this handyman. The handyman, in my opinion, is using a lightweight DIY snake that just isn't "horse enough" to do the job. Pro equipment is dangerous in the hands of an untrained person, so a real plumber should be brought in.

Thanks for the replies.
I will insist the super bring in a plumber with pro equipment, -longer, thicker, snake, remote camera?, etc.
I have no way of knowing if there is a baby upstairs.
I hope it's just adults getting ready for work, all showering / bathing at the same time.
I don't want to wrap my head around the idea that it is feces, even from a baby!